"""
DaoMath Relativity Theory (道数相对论)
从 ±0 先天均衡构造时空与引力

相对论的 ±0 诠释:
    时间 ↔ 空间 (对偶)
    物质 ↔ 时空 (对偶)
    收缩 ↔ 膨胀 (对偶)
    
理论层级:
    ±0 (先天均衡)
     ↓
    时空流形 (Spacetime Manifold)
     ↓
    度规张量 (Metric Tensor)
     ↓
    测地线 (Geodesics)
     ↓
    曲率张量 (Riemann Curvature)
     ↓
    Einstein 场方程
     ↓
    引力波与黑洞
"""

from __future__ import annotations
from typing import (
    List, Tuple, Callable, Optional, Dict, Any,
    Union, TypeVar
)
from dataclasses import dataclass, field
from abc import ABC, abstractmethod
from enum import Enum
import numpy as np
import scipy.linalg as la
from scipy.integrate import odeint, solve_ivp
from scipy.optimize import fsolve
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

from core.axioms import InnateBalance, DualPair
from structures.space import Manifold


# ============================================================================
# 基础常数
# ============================================================================

class PhysicalConstants:
    """物理常数"""
    C = 299792458.0              # 光速 (m/s)
    G = 6.67430e-11              # 引力常数 (m³/kg·s²)
    HBAR = 1.054571817e-34       # 约化普朗克常数 (J·s)
    K_B = 1.380649e-23           # 玻尔兹曼常数 (J/K)
    
    # 几何单位制: c = G = 1
    C_GEOM = 1.0
    G_GEOM = 1.0
    
    # 普朗克单位
    PLANCK_LENGTH = 1.616255e-35  # m
    PLANCK_TIME = 5.391247e-44    # s
    PLANCK_MASS = 2.176434e-8     # kg


# ============================================================================
# Level 1: 时空结构 (Spacetime Structure)
# ============================================================================

class SpacetimeCoordinates:
    """
    时空坐标系统
    
    (t, x, y, z) 或 (x⁰, x¹, x², x³)
    """
    
    def __init__(self, t: float, x: float, y: float, z: float):
        """
        Args:
            t: 时间坐标
            x, y, z: 空间坐标
        """
        self.t = t
        self.x = x
        self.y = y
        self.z = z
    
    def to_array(self) -> np.ndarray:
        """转换为数组"""
        return np.array([self.t, self.x, self.y, self.z])
    
    def __repr__(self) -> str:
        return f"({self.t:.3f}, {self.x:.3f}, {self.y:.3f}, {self.z:.3f})"
    
    @staticmethod
    def from_array(coords: np.ndarray) -> SpacetimeCoordinates:
        """从数组创建"""
        return SpacetimeCoordinates(coords[0], coords[1], coords[2], coords[3])


class SpacetimeInterval:
    """
    时空间隔
    
    基于 ±0 的理解:
    - 类时间隔 (ds² < 0): 时间主导
    - 类空间隔 (ds² > 0): 空间主导  
    - 类光间隔 (ds² = 0): 时空平衡
    """
    
    def __init__(self, metric: MetricTensor, p1: SpacetimeCoordinates, 
                 p2: SpacetimeCoordinates):
        """
        Args:
            metric: 度规张量
            p1, p2: 两个时空点
        """
        self.metric = metric
        self.p1 = p1
        self.p2 = p2
    
    def interval_squared(self) -> float:
        """
        间隔平方: ds² = g_μν dx^μ dx^ν
        """
        dx = self.p2.to_array() - self.p1.to_array()
        g = self.metric.at_point(self.p1.to_array())
        return float(dx @ g @ dx)
    
    def interval(self) -> float:
        """间隔 (带符号)"""
        ds2 = self.interval_squared()
        return np.sqrt(abs(ds2)) * np.sign(ds2)
    
    def is_timelike(self) -> bool:
        """是否为类时"""
        return self.interval_squared() < 0
    
    def is_spacelike(self) -> bool:
        """是否为类空"""
        return self.interval_squared() > 0
    
    def is_lightlike(self) -> bool:
        """是否为类光（零测地线）"""
        return abs(self.interval_squared()) < 1e-10
    
    def proper_time(self) -> Optional[float]:
        """
        固有时: τ = √(-ds²)
        
        只对类时间隔有意义
        """
        if self.is_timelike():
            return np.sqrt(-self.interval_squared())
        return None
    
    def proper_distance(self) -> Optional[float]:
        """
        固有距离: σ = √(ds²)
        
        只对类空间隔有意义
        """
        if self.is_spacelike():
            return np.sqrt(self.interval_squared())
        return None


# ============================================================================
# Level 2: 度规张量 (Metric Tensor)
# ============================================================================

class MetricTensor:
    """
    度规张量 g_μν
    
    定义时空的几何结构
    
    基于 ±0 的理解:
    - 时间分量 (g₀₀): 负号 (对偶于空间)
    - 空间分量 (gᵢⱼ): 正号
    - 非对角元: 时空混合
    """
    
    def __init__(self, metric_func: Callable[[np.ndarray], np.ndarray]):
        """
        Args:
            metric_func: 坐标 -> 4×4 度规矩阵的函数
        """
        self.metric_func = metric_func
    
    def at_point(self, coords: np.ndarray) -> np.ndarray:
        """在给定点计算度规"""
        return self.metric_func(coords)
    
    def inverse(self, coords: np.ndarray) -> np.ndarray:
        """逆度规 g^μν"""
        g = self.at_point(coords)
        return np.linalg.inv(g)
    
    def determinant(self, coords: np.ndarray) -> float:
        """度规行列式 det(g)"""
        g = self.at_point(coords)
        return float(np.linalg.det(g))
    
    def signature(self, coords: np.ndarray) -> Tuple[int, int]:
        """
        度规符号差
        
        Returns:
            (正本征值数, 负本征值数)
        """
        g = self.at_point(coords)
        eigenvalues = np.linalg.eigvalsh(g)
        positive = np.sum(eigenvalues > 1e-10)
        negative = np.sum(eigenvalues < -1e-10)
        return (positive, negative)
    
    def christoffel_symbols(self, coords: np.ndarray, 
                           delta: float = 1e-6) -> np.ndarray:
        """
        Christoffel 符号: Γ^μ_νρ = ½ g^μσ (∂_ν g_σρ + ∂_ρ g_νσ - ∂_σ g_νρ)
        
        使用数值微分计算
        """
        g = self.at_point(coords)
        g_inv = np.linalg.inv(g)
        
        Gamma = np.zeros((4, 4, 4))
        
        for mu in range(4):
            for nu in range(4):
                for rho in range(4):
                    sum_term = 0.0
                    
                    for sigma in range(4):
                        # ∂_ν g_σρ
                        coords_plus = coords.copy()
                        coords_plus[nu] += delta
                        g_plus = self.at_point(coords_plus)
                        dg_nu = (g_plus[sigma, rho] - g[sigma, rho]) / delta
                        
                        # ∂_ρ g_νσ
                        coords_plus = coords.copy()
                        coords_plus[rho] += delta
                        g_plus = self.at_point(coords_plus)
                        dg_rho = (g_plus[nu, sigma] - g[nu, sigma]) / delta
                        
                        # ∂_σ g_νρ
                        coords_plus = coords.copy()
                        coords_plus[sigma] += delta
                        g_plus = self.at_point(coords_plus)
                        dg_sigma = (g_plus[nu, rho] - g[nu, rho]) / delta
                        
                        sum_term += g_inv[mu, sigma] * (dg_nu + dg_rho - dg_sigma)
                    
                    Gamma[mu, nu, rho] = 0.5 * sum_term
        
        return Gamma
    
    @staticmethod
    def minkowski() -> MetricTensor:
        """
        Minkowski 度规 (平直时空)
        
        η_μν = diag(-1, +1, +1, +1)
        
        体现时空对偶: 时间(-) vs 空间(+)
        """
        def metric_func(coords: np.ndarray) -> np.ndarray:
            return np.diag([-1.0, 1.0, 1.0, 1.0])
        
        return MetricTensor(metric_func)
    
    @staticmethod
    def schwarzschild(mass: float) -> MetricTensor:
        """
        Schwarzschild 度规 (球对称引力场)
        
        ds² = -(1-2M/r)dt² + (1-2M/r)⁻¹dr² + r²(dθ² + sin²θ dφ²)
        
        体现引力的时空弯曲
        """
        def metric_func(coords: np.ndarray) -> np.ndarray:
            # coords = (t, r, θ, φ)
            t, r, theta, phi = coords
            
            # Schwarzschild 半径
            r_s = 2 * mass
            
            if r <= r_s:
                # 视界内部（需要特殊处理）
                r = r_s + 1e-6
            
            f = 1 - r_s / r
            
            g = np.zeros((4, 4))
            g[0, 0] = -f                    # g_tt
            g[1, 1] = 1 / f if f > 1e-10 else 1e10  # g_rr
            g[2, 2] = r**2                  # g_θθ
            g[3, 3] = r**2 * np.sin(theta)**2  # g_φφ
            
            return g
        
        return MetricTensor(metric_func)
    
    @staticmethod
    def kerr(mass: float, angular_momentum: float) -> MetricTensor:
        """
        Kerr 度规 (旋转黑洞)
        
        包含参数:
        - M: 质量
        - a = J/M: 比角动量
        """
        a = angular_momentum / mass if mass > 0 else 0
        
        def metric_func(coords: np.ndarray) -> np.ndarray:
            t, r, theta, phi = coords
            
            r_s = 2 * mass
            
            # Boyer-Lindquist 坐标
            Sigma = r**2 + a**2 * np.cos(theta)**2
            Delta = r**2 - r_s * r + a**2
            A = (r**2 + a**2)**2 - a**2 * Delta * np.sin(theta)**2
            
            g = np.zeros((4, 4))
            
            # 对角元
            g[0, 0] = -(1 - r_s * r / Sigma)
            g[1, 1] = Sigma / Delta if Delta > 1e-10 else 1e10
            g[2, 2] = Sigma
            g[3, 3] = A * np.sin(theta)**2 / Sigma
            
            # 非对角元（时空拖曳）
            g[0, 3] = -r_s * r * a * np.sin(theta)**2 / Sigma
            g[3, 0] = g[0, 3]
            
            return g
        
        return MetricTensor(metric_func)
    
    @staticmethod
    def flrw(scale_factor: Callable[[float], float], 
             curvature: float = 0) -> MetricTensor:
        """
        FLRW 度规 (宇宙学)
        
        ds² = -dt² + a(t)²[dr²/(1-kr²) + r²(dθ² + sin²θ dφ²)]
        
        Args:
            scale_factor: 尺度因子 a(t)
            curvature: 空间曲率 k = -1, 0, +1
        """
        def metric_func(coords: np.ndarray) -> np.ndarray:
            t, r, theta, phi = coords
            a_t = scale_factor(t)
            k = curvature
            
            g = np.zeros((4, 4))
            g[0, 0] = -1.0
            g[1, 1] = a_t**2 / (1 - k * r**2) if abs(1 - k * r**2) > 1e-10 else a_t**2 * 1e10
            g[2, 2] = a_t**2 * r**2
            g[3, 3] = a_t**2 * r**2 * np.sin(theta)**2
            
            return g
        
        return MetricTensor(metric_func)


# ============================================================================
# Level 3: 测地线 (Geodesics)
# ============================================================================

class Geodesic:
    """
    测地线: 时空中的"直线"
    
    满足测地线方程:
    d²x^μ/dλ² + Γ^μ_νρ (dx^ν/dλ)(dx^ρ/dλ) = 0
    
    基于 ±0 的理解:
    - 测地线是极值路径（能量-动量均衡）
    - 自由落体遵循测地线（引力几何化）
    """
    
    def __init__(self, metric: MetricTensor):
        self.metric = metric
    
    def equation_of_motion(self, 
                          state: np.ndarray, 
                          lambda_param: float) -> np.ndarray:
        """
        测地线方程的一阶形式
        
        状态: [x⁰, x¹, x², x³, u⁰, u¹, u², u³]
        其中 u^μ = dx^μ/dλ
        """
        coords = state[:4]
        velocity = state[4:]
        
        # 计算 Christoffel 符号
        Gamma = self.metric.christoffel_symbols(coords)
        
        # 加速度: a^μ = -Γ^μ_νρ u^ν u^ρ
        acceleration = np.zeros(4)
        for mu in range(4):
            for nu in range(4):
                for rho in range(4):
                    acceleration[mu] -= Gamma[mu, nu, rho] * velocity[nu] * velocity[rho]
        
        # 返回导数: [u^μ, a^μ]
        return np.concatenate([velocity, acceleration])
    
    def integrate(self,
                 initial_position: np.ndarray,
                 initial_velocity: np.ndarray,
                 lambda_span: Tuple[float, float],
                 num_points: int = 100) -> Tuple[np.ndarray, np.ndarray]:
        """
        积分测地线方程
        
        Returns:
            lambda_values: 参数值
            trajectory: 轨迹 [x^μ(λ), u^μ(λ)]
        """
        initial_state = np.concatenate([initial_position, initial_velocity])
        lambda_values = np.linspace(lambda_span[0], lambda_span[1], num_points)
        
        solution = odeint(
            self.equation_of_motion,
            initial_state,
            lambda_values
        )
        
        return lambda_values, solution
    
    def proper_time_parametrization(self, trajectory: np.ndarray) -> np.ndarray:
        """
        计算固有时参数化
        
        dτ² = -g_μν dx^μ dx^ν
        """
        proper_times = [0.0]
        
        for i in range(1, len(trajectory)):
            coords1 = trajectory[i-1, :4]
            coords2 = trajectory[i, :4]
            
            dx = coords2 - coords1
            g = self.metric.at_point(coords1)
            ds2 = dx @ g @ dx
            
            if ds2 < 0:  # 类时
                dtau = np.sqrt(-ds2)
            else:
                dtau = 0.0
            
            proper_times.append(proper_times[-1] + dtau)
        
        return np.array(proper_times)


class NullGeodesic(Geodesic):
    """
    零测地线（光线）
    
    满足: ds² = 0
    """
    
    def __init__(self, metric: MetricTensor):
        super().__init__(metric)
    
    def verify_null_condition(self, trajectory: np.ndarray) -> bool:
        """验证是否满足零测地线条件"""
        for coords, velocity in zip(trajectory[:, :4], trajectory[:, 4:]):
            g = self.metric.at_point(coords)
            ds2 = velocity @ g @ velocity
            
            if abs(ds2) > 1e-6:
                return False
        
        return True


# ============================================================================
# Level 4: 曲率张量 (Curvature Tensor)
# ============================================================================

class RiemannCurvature:
    """
    Riemann 曲率张量 R^μ_νρσ
    
    度量时空弯曲程度
    
    基于 ±0 的理解:
    - 曲率是物质-能量与时空几何的均衡
    - Ricci 曲率: 体积变化
    - Weyl 曲率: 形状变化
    """
    
    def __init__(self, metric: MetricTensor):
        self.metric = metric
    
    def riemann_tensor(self, coords: np.ndarray, 
                      delta: float = 1e-5) -> np.ndarray:
        """
        Riemann 曲率张量
        
        R^μ_νρσ = ∂_ρ Γ^μ_νσ - ∂_σ Γ^μ_νρ + Γ^μ_λρ Γ^λ_νσ - Γ^μ_λσ Γ^λ_νρ
        """
        Gamma = self.metric.christoffel_symbols(coords)
        R = np.zeros((4, 4, 4, 4))
        
        for mu in range(4):
            for nu in range(4):
                for rho in range(4):
                    for sigma in range(4):
                        # ∂_ρ Γ^μ_νσ
                        coords_plus = coords.copy()
                        coords_plus[rho] += delta
                        Gamma_plus = self.metric.christoffel_symbols(coords_plus)
                        dGamma_rho = (Gamma_plus[mu, nu, sigma] - Gamma[mu, nu, sigma]) / delta
                        
                        # ∂_σ Γ^μ_νρ
                        coords_plus = coords.copy()
                        coords_plus[sigma] += delta
                        Gamma_plus = self.metric.christoffel_symbols(coords_plus)
                        dGamma_sigma = (Gamma_plus[mu, nu, rho] - Gamma[mu, nu, rho]) / delta
                        
                        # Γ^μ_λρ Γ^λ_νσ
                        term1 = sum(Gamma[mu, lam, rho] * Gamma[lam, nu, sigma] 
                                   for lam in range(4))
                        
                        # Γ^μ_λσ Γ^λ_νρ
                        term2 = sum(Gamma[mu, lam, sigma] * Gamma[lam, nu, rho] 
                                   for lam in range(4))
                        
                        R[mu, nu, rho, sigma] = dGamma_rho - dGamma_sigma + term1 - term2
        
        return R
    
    def ricci_tensor(self, coords: np.ndarray) -> np.ndarray:
        """
        Ricci 曲率张量: R_μν = R^ρ_μρν
        """
        R = self.riemann_tensor(coords)
        Ric = np.zeros((4, 4))
        
        for mu in range(4):
            for nu in range(4):
                Ric[mu, nu] = sum(R[rho, mu, rho, nu] for rho in range(4))
        
        return Ric
    
    def ricci_scalar(self, coords: np.ndarray) -> float:
        """
        Ricci 标量曲率: R = g^μν R_μν
        """
        Ric = self.ricci_tensor(coords)
        g_inv = self.metric.inverse(coords)
        
        R_scalar = sum(g_inv[mu, nu] * Ric[mu, nu] 
                      for mu in range(4) for nu in range(4))
        
        return float(R_scalar)
    
    def einstein_tensor(self, coords: np.ndarray) -> np.ndarray:
        """
        Einstein 张量: G_μν = R_μν - ½ g_μν R
        """
        Ric = self.ricci_tensor(coords)
        R_scalar = self.ricci_scalar(coords)
        g = self.metric.at_point(coords)
        
        G = Ric - 0.5 * R_scalar * g
        
        return G
    
    def kretschmann_scalar(self, coords: np.ndarray) -> float:
        """
        Kretschmann 标量: K = R_μνρσ R^μνρσ
        
        曲率不变量，用于检测时空奇点
        """
        R = self.riemann_tensor(coords)
        g_inv = self.metric.inverse(coords)
        
        K = 0.0
        for mu in range(4):
            for nu in range(4):
                for rho in range(4):
                    for sigma in range(4):
                        for alpha in range(4):
                            for beta in range(4):
                                for gamma in range(4):
                                    for delta in range(4):
                                        K += (R[mu, nu, rho, sigma] * 
                                             g_inv[mu, alpha] * g_inv[nu, beta] * 
                                             g_inv[rho, gamma] * g_inv[sigma, delta] * 
                                             R[alpha, beta, gamma, delta])
        
        return float(K)


# ============================================================================
# Level 5: Einstein 场方程 (Einstein Field Equations)
# ============================================================================

class EinsteinFieldEquations:
    """
    Einstein 场方程
    
    G_μν + Λg_μν = (8πG/c⁴) T_μν
    
    基于 ±0 的理解:
    - 左边: 几何（时空曲率）
    - 右边: 物质（能量-动量）
    - 方程: 几何-物质的均衡
    """
    
    def __init__(self, 
                 metric: MetricTensor,
                 cosmological_constant: float = 0.0):
        """
        Args:
            metric: 度规张量
            cosmological_constant: 宇宙学常数 Λ
        """
        self.metric = metric
        self.Lambda = cosmological_constant
        self.curvature = RiemannCurvature(metric)
    
    def field_equation_lhs(self, coords: np.ndarray) -> np.ndarray:
        """
        场方程左边: G_μν + Λg_μν
        """
        G = self.curvature.einstein_tensor(coords)
        g = self.metric.at_point(coords)
        
        return G + self.Lambda * g
    
    def stress_energy_from_field(self, coords: np.ndarray) -> np.ndarray:
        """
        从场方程反推能量-动量张量
        
        T_μν = (c⁴/8πG)(G_μν + Λg_μν)
        """
        lhs = self.field_equation_lhs(coords)
        
        c4_over_8piG = (PhysicalConstants.C**4) / (8 * np.pi * PhysicalConstants.G)
        
        T = c4_over_8piG * lhs
        
        return T
    
    def verify_field_equations(self,
                               coords: np.ndarray,
                               stress_energy: np.ndarray,
                               tolerance: float = 1e-6) -> bool:
        """
        验证场方程是否满足
        """
        lhs = self.field_equation_lhs(coords)
        
        c4_over_8piG = (PhysicalConstants.C**4) / (8 * np.pi * PhysicalConstants.G)
        rhs = c4_over_8piG * stress_energy
        
        difference = np.abs(lhs - rhs)
        
        return np.all(difference < tolerance)
    
    @staticmethod
    def perfect_fluid_stress_energy(
        density: float,
        pressure: float,
        four_velocity: np.ndarray,
        metric: np.ndarray
    ) -> np.ndarray:
        """
        完美流体的能量-动量张量
        
        T_μν = (ρ + p/c²) u_μ u_ν + p g_μν
        
        Args:
            density: 能量密度 ρ
            pressure: 压强 p
            four_velocity: 四速度 u^μ
            metric: 度规 g_μν
        """
        # 降指标: u_μ = g_μν u^ν
        u_lower = metric @ four_velocity
        
        # T_μν = (ρ + p) u_μ u_ν + p g_μν
        T = (density + pressure) * np.outer(u_lower, u_lower) + pressure * metric
        
        return T
    
    @staticmethod
    def electromagnetic_stress_energy(
        electric_field: np.ndarray,
        magnetic_field: np.ndarray
    ) -> np.ndarray:
        """
        电磁场的能量-动量张量
        
        T^μν = (1/μ₀)[F^μ_ρ F^νρ - ¼ g^μν F_ρσ F^ρσ]
        """
        # 简化实现：仅考虑能量密度
        epsilon_0 = 8.854187817e-12  # 真空介电常数
        
        E = electric_field
        B = magnetic_field
        
        # 能量密度
        energy_density = 0.5 * epsilon_0 * (np.dot(E, E) + 
                                            np.dot(B, B) / (4 * np.pi * 1e-7))
        
        T = np.zeros((4, 4))
        T[0, 0] = energy_density
        
        # 动量密度和应力张量（简化）
        poynting = epsilon_0 * np.cross(E, B)
        for i in range(3):
            T[0, i+1] = poynting[i]
            T[i+1, 0] = poynting[i]
        
        return T


# ============================================================================
# Level 6: 特殊解与物理现象
# ============================================================================

class SchwarzschildSolution:
    """
    Schwarzschild 解
    
    球对称、静态、真空解
    描述非旋转黑洞
    """
    
    def __init__(self, mass: float):
        """
        Args:
            mass: 黑洞质量（几何单位）
        """
        self.mass = mass
        self.r_s = 2 * mass  # Schwarzschild 半径
        self.metric = MetricTensor.schwarzschild(mass)
    
    def event_horizon_radius(self) -> float:
        """事件视界半径"""
        return self.r_s
    
    def photon_sphere_radius(self) -> float:
        """光子球半径: r = 3M"""
        return 3 * self.mass
    
    def innermost_stable_circular_orbit(self) -> float:
        """最内稳定圆轨道 (ISCO): r = 6M"""
        return 6 * self.mass
    
    def schwarzschild_time(self, coordinate_time: float, radius: float) -> float:
        """
        Schwarzschild 时间 vs 坐标时间
        
        dt_shell = dt_inf √(1 - r_s/r)
        """
        if radius <= self.r_s:
            return 0.0
        
        time_dilation = np.sqrt(1 - self.r_s / radius)
        return coordinate_time * time_dilation
    
    def effective_potential(self, radius: float, angular_momentum: float) -> float:
        """
        有效势能（径向运动）
        
        V_eff = (1 - r_s/r)(1 + L²/r²)
        """
        if radius <= 0:
            return float('inf')
        
        return (1 - self.r_s / radius) * (1 + angular_momentum**2 / radius**2)
    
    def orbital_velocity(self, radius: float) -> float:
        """
        圆轨道速度
        
        v = √(M/r)
        """
        if radius <= self.r_s:
            return 1.0  # 光速（几何单位）
        
        return np.sqrt(self.mass / radius)
    
    def hawking_temperature(self) -> float:
        """
        Hawking 温度
        
        T_H = ℏc³/(8πGMk_B)
        """
        hbar = PhysicalConstants.HBAR
        c = PhysicalConstants.C
        G = PhysicalConstants.G
        k_B = PhysicalConstants.K_B
        
        return (hbar * c**3) / (8 * np.pi * G * self.mass * k_B)
    
    def bekenstein_hawking_entropy(self) -> float:
        """
        Bekenstein-Hawking 熵
        
        S = (k_B c³ A)/(4Gℏ) = k_B (4πM²)
        
        其中 A = 4πr_s² 是视界面积
        """
        k_B = PhysicalConstants.K_B
        area = 4 * np.pi * self.r_s**2
        
        return k_B * area / 4


class KerrSolution:
    """
    Kerr 解
    
    轴对称、稳态、旋转黑洞
    """
    
    def __init__(self, mass: float, angular_momentum: float):
        """
        Args:
            mass: 黑洞质量
            angular_momentum: 角动量 J
        """
        self.mass = mass
        self.angular_momentum = angular_momentum
        self.a = angular_momentum / mass if mass > 0 else 0  # 比角动量
        
        self.metric = MetricTensor.kerr(mass, angular_momentum)
    
    def outer_horizon_radius(self) -> float:
        """外视界半径: r₊ = M + √(M² - a²)"""
        M = self.mass
        discriminant = M**2 - self.a**2
        
        if discriminant < 0:
            # 超极端 Kerr (不物理)
            return M
        
        return M + np.sqrt(discriminant)
    
    def inner_horizon_radius(self) -> float:
        """内视界半径: r₋ = M - √(M² - a²)"""
        M = self.mass
        discriminant = M**2 - self.a**2
        
        if discriminant < 0:
            return M
        
        return M - np.sqrt(discriminant)
    
    def ergosphere_radius(self, theta: float) -> float:
        """
        能层半径
        
        r_ergo = M + √(M² - a² cos²θ)
        """
        M = self.mass
        discriminant = M**2 - self.a**2 * np.cos(theta)**2
        
        return M + np.sqrt(max(0, discriminant))
    
    def frame_dragging_angular_velocity(self, radius: float, theta: float) -> float:
        """
        参考系拖曳角速度（Lense-Thirring 效应）
        
        ω = 2Mar/(r⁴ + a²r² + 2Ma²r)
        """
        M = self.mass
        a = self.a
        
        numerator = 2 * M * a * radius
        denominator = radius**4 + a**2 * radius**2 + 2 * M * a**2 * radius
        
        if denominator == 0:
            return 0.0
        
        return numerator / denominator
    
    def penrose_process_efficiency(self) -> float:
        """
        Penrose 过程的最大效率
        
        η = 1 - √(1 - a²/M²) ≤ 29%
        """
        discriminant = 1 - (self.a / self.mass)**2
        
        if discriminant < 0:
            return 0.29  # 极端情况
        
        return 1 - np.sqrt(discriminant)


class FriedmannUniverse:
    """
    Friedmann 宇宙学模型
    
    均匀、各向同性的膨胀宇宙
    """
    
    def __init__(self, 
                 H0: float,
                 Omega_m: float,
                 Omega_Lambda: float,
                 curvature: float = 0):
        """
        Args:
            H0: 当前 Hubble 常数
            Omega_m: 物质密度参数
            Omega_Lambda: 暗能量密度参数
            curvature: 空间曲率 k
        """
        self.H0 = H0
        self.Omega_m = Omega_m
        self.Omega_Lambda = Omega_Lambda
        self.Omega_k = 1 - Omega_m - Omega_Lambda
        self.curvature = curvature
    
    def scale_factor(self, time: float) -> float:
        """
        尺度因子 a(t)
        
        通过 Friedmann 方程求解
        """
        # 简化：假设物质主导
        if self.Omega_m > 0.5:
            # a ∝ t^(2/3)
            return (time / 1.0)**(2/3)
        elif self.Omega_Lambda > 0.5:
            # a ∝ e^(Ht)
            return np.exp(self.H0 * time)
        else:
            # 线性增长（辐射主导）
            return time**0.5
    
    def hubble_parameter(self, time: float) -> float:
        """
        Hubble 参数 H(t) = ȧ/a
        """
        dt = 1e-6
        a_now = self.scale_factor(time)
        a_future = self.scale_factor(time + dt)
        
        da_dt = (a_future - a_now) / dt
        
        return da_dt / a_now if a_now > 0 else 0
    
    def friedmann_equation(self, scale_factor: float) -> float:
        """
        第一 Friedmann 方程
        
        H² = (8πG/3)ρ - k/a² + Λ/3
        """
        a = scale_factor
        
        H_squared = (self.Omega_m / a**3 + 
                    self.Omega_k / a**2 + 
                    self.Omega_Lambda) * self.H0**2
        
        return np.sqrt(max(0, H_squared))
    
    def age_of_universe(self) -> float:
        """
        宇宙年龄 t₀
        
        t₀ = ∫₀¹ da/(aH(a))
        """
        def integrand(a):
            if a <= 0:
                return 0
            H = self.friedmann_equation(a)
            return 1 / (a * H) if H > 0 else 0
        
        from scipy.integrate import quad
        age, _ = quad(integrand, 1e-10, 1.0)
        
        return age
    
    def critical_density(self) -> float:
        """
        临界密度 ρ_c = 3H₀²/(8πG)
        """
        return 3 * self.H0**2 / (8 * np.pi * PhysicalConstants.G)


# ============================================================================
# Level 7: 引力波 (Gravitational Waves)
# ============================================================================

class GravitationalWave:
    """
    引力波
    
    度规微扰: g_μν = η_μν + h_μν
    
    基于 ±0 的理解:
    - h₊: 加号极化（正向）
    - h_× : 叉号极化（负向）
    - 两者对偶
    """
    
    def __init__(self, 
                 amplitude: float,
                 frequency: float,
                 polarization: str = 'plus'):
        """
        Args:
            amplitude: 振幅 h₀
            frequency: 频率 f
            polarization: 极化模式 ('plus' 或 'cross')
        """
        self.amplitude = amplitude
        self.frequency = frequency
        self.omega = 2 * np.pi * frequency
        self.polarization = polarization
    
    def strain_tensor(self, t: float, x: float, y: float, z: float) -> np.ndarray:
        """
        应变张量 h_μν(t, x)
        
        平面波解: h_μν = A_μν e^(i(kz - ωt))
        """
        phase = self.omega * (t - z / PhysicalConstants.C)
        h0 = self.amplitude * np.cos(phase)
        
        h = np.zeros((4, 4))
        
        if self.polarization == 'plus':
            # h₊ 极化
            h[1, 1] = h0      # h_xx
            h[2, 2] = -h0     # h_yy
        elif self.polarization == 'cross':
            # h_× 极化
            h[1, 2] = h0      # h_xy
            h[2, 1] = h0      # h_yx
        
        return h
    
    def tidal_force(self, separation: np.ndarray, time: float) -> np.ndarray:
        """
        潮汐力（对测试粒子的影响）
        
        F^i ∝ h^i_j x^j
        """
        x, y, z = separation
        h = self.strain_tensor(time, x, y, z)
        
        tidal = np.array([
            h[1, 1] * x + h[1, 2] * y,
            h[2, 1] * x + h[2, 2] * y,
            0.0  # 沿传播方向无力
        ])
        
        return tidal
    
    def energy_flux(self) -> float:
        """
        能量通量 (per unit area)
        
        dE/dA dt = (c³/16πG) ⟨ḣ²_μν⟩
        """
        c = PhysicalConstants.C
        G = PhysicalConstants.G
        
        # 时间平均
        h_dot_squared = (self.amplitude * self.omega)**2 / 2
        
        flux = (c**3 / (16 * np.pi * G)) * h_dot_squared
        
        return flux
    
    def characteristic_strain(self, distance: float) -> float:
        """
        特征应变
        
        h_c = √(2E_GW/(πM c² f))
        """
        # 简化估计
        return self.amplitude / distance
    
    @staticmethod
    def from_binary_system(
        mass1: float,
        mass2: float,
        orbital_frequency: float,
        distance: float
    ) -> GravitationalWave:
        """
        从双星系统计算引力波
        
        Args:
            mass1, mass2: 双星质量
            orbital_frequency: 轨道频率
            distance: 到观测者的距离
        """
        # Chirp 质量
        M_chirp = (mass1 * mass2)**(3/5) / (mass1 + mass2)**(1/5)
        
        # 引力波频率 = 2 × 轨道频率
        f_GW = 2 * orbital_frequency
        
        # 振幅估计（简化）
        G = PhysicalConstants.G
        c = PhysicalConstants.C
        
        amplitude = (4 * G * M_chirp / c**2) * (np.pi * f_GW * G * M_chirp / c**3)**(2/3) / distance
        
        return GravitationalWave(amplitude, f_GW, 'plus')


# ============================================================================
# 综合演示系统
# ============================================================================

class RelativityDemo:
    """相对论综合演示"""
    
    @staticmethod
    def demonstrate_spacetime_interval():
        """演示时空间隔"""
        print("\n" + "=" * 70)
        print("时空间隔演示")
        print("=" * 70)
        
        metric = MetricTensor.minkowski()
        
        # 三种间隔类型
        p0 = SpacetimeCoordinates(0, 0, 0, 0)
        
        # 类时间隔
        p1_timelike = SpacetimeCoordinates(1, 0.5, 0, 0)
        interval1 = SpacetimeInterval(metric, p0, p1_timelike)
        
        print("\n类时间隔 (时间主导):")
        print(f"  p0 = {p0}")
        print(f"  p1 = {p1_timelike}")
        print(f"  ds² = {interval1.interval_squared():.4f} < 0")
        print(f"  固有时 τ = {interval1.proper_time():.4f}")
        
        # 类空间隔
        p2_spacelike = SpacetimeCoordinates(0.5, 1, 0, 0)
        interval2 = SpacetimeInterval(metric, p0, p2_spacelike)
        
        print("\n类空间隔 (空间主导):")
        print(f"  p0 = {p0}")
        print(f"  p2 = {p2_spacelike}")
        print(f"  ds² = {interval2.interval_squared():.4f} > 0")
        print(f"  固有距离 σ = {interval2.proper_distance():.4f}")
        
        # 类光间隔
        p3_lightlike = SpacetimeCoordinates(1, 1, 0, 0)
        interval3 = SpacetimeInterval(metric, p0, p3_lightlike)
        
        print("\n类光间隔 (光速传播):")
        print(f"  p0 = {p0}")
        print(f"  p3 = {p3_lightlike}")
        print(f"  ds² = {interval3.interval_squared():.6f} ≈ 0")
        print(f"  是类光: {interval3.is_lightlike()}")
    
    @staticmethod
    def demonstrate_schwarzschild_geodesics():
        """演示 Schwarzschild 测地线"""
        print("\n" + "=" * 70)
        print("Schwarzschild 黑洞测地线")
        print("=" * 70)
        
        mass = 1.0  # 几何单位
        schwarzschild = SchwarzschildSolution(mass)
        
        print(f"\n黑洞质量 M = {mass}")
        print(f"事件视界半径 r_s = {schwarzschild.event_horizon_radius():.2f}")
        print(f"光子球半径 r_ph = {schwarzschild.photon_sphere_radius():.2f}")
        print(f"ISCO半径 r_isco = {schwarzschild.innermost_stable_circular_orbit():.2f}")
        
        # 圆轨道速度
        radii = [4, 6, 10, 20]
        print("\n圆轨道速度:")
        for r in radii:
            v = schwarzschild.orbital_velocity(r)
            print(f"  r = {r:2.0f}M: v = {v:.4f}c")
        
        # Hawking 辐射
        T_H = schwarzschild.hawking_temperature()
        print(f"\nHawking 温度: T_H = {T_H:.2e} K")
        
        S_BH = schwarzschild.bekenstein_hawking_entropy()
        print(f"Bekenstein-Hawking 熵: S = {S_BH:.2e}")
    
    @staticmethod
    def demonstrate_kerr_black_hole():
        """演示 Kerr 旋转黑洞"""
        print("\n" + "=" * 70)
        print("Kerr 旋转黑洞")
        print("=" * 70)
        
        mass = 1.0
        angular_momentum = 0.8  # 快速旋转
        
        kerr = KerrSolution(mass, angular_momentum)
        
        print(f"\n黑洞参数:")
        print(f"  质量 M = {mass}")
        print(f"  角动量 J = {angular_momentum}")
        print(f"  比角动量 a = {kerr.a:.2f}")
        
        print(f"\n视界半径:")
        print(f"  外视界 r₊ = {kerr.outer_horizon_radius():.4f}")
        print(f"  内视界 r₋ = {kerr.inner_horizon_radius():.4f}")
        
        # 能层
        print(f"\n能层半径 (赤道):")
        r_ergo = kerr.ergosphere_radius(np.pi/2)
        print(f"  r_ergo = {r_ergo:.4f}")
        
        # Penrose 过程
        efficiency = kerr.penrose_process_efficiency()
        print(f"\nPenrose 过程最大效率: {efficiency*100:.1f}%")
    
    @staticmethod
    def demonstrate_cosmology():
        """演示宇宙学模型"""
        print("\n" + "=" * 70)
        print("Friedmann 宇宙学")
        print("=" * 70)
        
        # 标准 ΛCDM 模型参数
        H0 = 70  # km/s/Mpc (简化单位)
        Omega_m = 0.3
        Omega_Lambda = 0.7
        
        universe = FriedmannUniverse(H0, Omega_m, Omega_Lambda)
        
        print(f"\n宇宙参数:")
        print(f"  H₀ = {H0} km/s/Mpc")
        print(f"  Ω_m = {Omega_m} (物质)")
        print(f"  Ω_Λ = {Omega_Lambda} (暗能量)")
        print(f"  Ω_k = {universe.Omega_k} (曲率)")
        
        # 临界密度
        rho_c = universe.critical_density()
        print(f"\n临界密度 ρ_c = {rho_c:.2e} kg/m³")
        
        # 宇宙年龄（简化计算）
        print(f"\n宇宙演化:")
        for a in [0.5, 0.75, 1.0]:
            H = universe.friedmann_equation(a)
            print(f"  尺度因子 a={a:.2f}: H={H:.2f} H₀")
    
    @staticmethod
    def demonstrate_gravitational_waves():
        """演示引力波"""
        print("\n" + "=" * 70)
        print("引力波")
        print("=" * 70)
        
        # 双黑洞系统
        mass1 = 30  # 太阳质量
        mass2 = 30
        f_orbital = 100  # Hz
        distance = 1e9  # 光年（简化）
        
        gw = GravitationalWave.from_binary_system(
            mass1, mass2, f_orbital, distance
        )
        
        print(f"\n双黑洞系统:")
        print(f"  质量: {mass1} M_☉ + {mass2} M_☉")
        print(f"  轨道频率: {f_orbital} Hz")
        print(f"  距离: {distance:.1e} 光年")
        
        print(f"\n引力波特性:")
        print(f"  频率: {gw.frequency} Hz")
        print(f"  振幅: {gw.amplitude:.2e}")
        print(f"  极化: {gw.polarization}")
        
        # 能量通量
        flux = gw.energy_flux()
        print(f"\n能量通量: {flux:.2e} W/m²")
        
        # 应变
        h_c = gw.characteristic_strain(distance)
        print(f"特征应变: {h_c:.2e}")


def main():
    """主函数：完整演示"""
    
    print("=" * 70)
    print("DaoMath Relativity Theory")
    print("从 ±0 构造时空与引力")
    print("=" * 70)
    
    # 1. 时空间隔
    RelativityDemo.demonstrate_spacetime_interval()
    
    # 2. Schwarzschild 黑洞
    RelativityDemo.demonstrate_schwarzschild_geodesics()
    
    # 3. Kerr 黑洞
    RelativityDemo.demonstrate_kerr_black_hole()
    
    # 4. 宇宙学
    RelativityDemo.demonstrate_cosmology()
    
    # 5. 引力波
    RelativityDemo.demonstrate_gravitational_waves()
    
    print("\n" + "=" * 70)
    print("相对论理论构造完成！")
    print("=" * 70)


if __name__ == "__main__":
    main()
